On some diophantine inequalities involving primes

2) 
Added to final draft: B y means of Linnik’s theorem on the smallest prime in an arithmetical progression, 
the stronger inequality \oc — qlp | < p ~ l ~6 can be established, where ô denotes a positive absolute constant.

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168
B a k e r , On some diophantine inequalities involving primes
The highest common factor of integers ra, n, . .
k  
is denoted by (m, ra, . . ., 
k ) .  
For each pair of integers m , q >  0 we define
and put
Cq(m) = J S  
3)
 = J ( 17 
C?(m) (?(? ))
i = i u - i
The series converges, as will be clear later from the proof of Lemma 7.
By c19 c2, . . . we shall mean positive absolute constants unless the contrary is 
specifically stated, [x] denotes the integral p a rt of x and || x || denotes th e distance of x 
from the nearest integer taken positively.
3. Lemmas
We now give fourteen lemmas prelim inary to the proof of th e theorem.
Lemma 1. For each integer 
b 
4
= 
0 and each 
a
on 
W a Q 
we have
(6) 
I 
S(bcc) - I(bfi)CQ(b)(cpM) - 1 1 < c, I 
!
where fi — a — a/q and cly c2 > 0 depend only on H .
Proof. The proof follows as in [4] (see p. 188) on noting th a t the conclusion of 
Lem m a 4 of [4] can be sharpened to
: 
*
\ f
h i ' 
} 
i 
¿ t ip ‘ h iIt
where c3y cx > 0 depend only on oc (see [3], Satz 8. 3, p. 144). The presence of | 
b 
| on 
the right of (6) results from the inequalities
N \ 1 — e(bfi) | ^ 2 j t N  | 
bf i
2
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